How do I split a face in half?
While looking around I found this answer that talks about splitting an edge with Subdivide and thought I'd try it with two opposite edges of a face selected. Turns out it works!

First, click on Edge select:



Then, select the top and bottom edges, since we want to split the face horizontally. If you want to split the face vertically, select the left and right edges instead.



Right click somewhere in the scene (don't think it matters where) and select Subdivide from the menu:



Ta-da! It's split perfectly in two.



Now you can select each face:



I'd be interested if anyone has a better way of doing this, as it seems like a very common task.